Existence and non-existence of transition fronts for bistable and ignition reactions
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Publication:1679682
DOI10.1016/J.ANIHPC.2016.11.004zbMATH Open1379.35165arXiv1503.07599OpenAlexW2503869908MaRDI QIDQ1679682
Author name not available (Why is that?)
Publication date: 21 November 2017
Published in: (Search for Journal in Brave)
Abstract: We study reaction-diffusion equations in one spatial dimension and with general (space- or time-) inhomogeneous mixed bistable-ignition reactions. For those satisfying a simple quantitative hypothesis, we prove existence and uniqueness of transition fronts, as well as convergence of "typical" solutions to the unique transition front (the existence part even extends to mixed bistable-ignition-monostable reactions). These results also hold for all pure ignition reactions without any other hypotheses, but not for all pure bistable reactions. In fact, we find examples of either spatially or temporally periodic pure bistable reactions (independent of the other space-time variable) for which we can prove non-existence of transition fronts. These are the first such results for periodic media which are non-degenerate in a natural sense, and the spatially periodic examples also prove a conjecture from cite{DHZ}.
Full work available at URL: https://arxiv.org/abs/1503.07599
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